After having used my 18" bino-Dobson for over a year now and having compared it extensively with other instruments, including a 27" mono-Dob, I can confirm that a 18" bino is at least equal, if not superior to a 25" mono. On faint objects I was even able to discern more detail than in the 27". Not only because the binocular summation factor is 1,42 (and not 1,18 as stated above), the increased contrast, more relaxed observing experience and reduced signal to noise ratio increase the summation factor to even 1,8x on those objects. I just have to close one eye to see the difference with a 18" mono and that difference is HUGE. If it were only 1,18x that difference would hardly be visible.
Also I'd like to clarify a misunderstanding. Counting in shipping and customs I paid significantly LESS for my binos than I would have paid for a 25" Obsession. Then again, double eyepieces and filters add to the expense...
Yes, the binocular summation factor is indeed 1.414. Signal to noise is increased by this ratio. You can see point sources 1.414 times fainter. You can discriminate subtle contrast 1.414 times better. Which equates to a mono aperture 1.189 times larger.
You can verify this by seeing how much fainter a star you can pick out. Theory predicts 0.37 magnitude, which is a tad better than most folks actually obtain. By characterizing the equivalent aperture ratio of factor 1.414 you are in effect claiming 0.75 magnitude, which would be astonishing to say the least.